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I'm looking to understand the practical details of calibrating local volatility to option prices for a range of different expiries using the Dupire local volatility equation. Would appreciate some guidance on the following:
a) Is there a good reference that outlines the key details of local vol calibration? Having n... |
Old and golden question, and maybe a new perspective:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3783233 |
Generally, a trading strategy is effective for a certain duration which is a function of the environment, news etc. I am currently learning about basic strategies that are (or once were) used. Some include simple mean reversion or standard deviation approaches others use more complex methods like machine learning for e... |
Could anyone show me how to get the second row equation from the first row equation please? For each letter, p is the risk-neutral probability in the risk-neutral world, u is the up factor for the stock, and d is the down factor for the stock, S0 is the beginning stock price. The equation is based on a one-step binomia... |
Old and golden question, and maybe a new perspective:
As the previous answers have pointed out, distinction needs to made between "skewness" and "skew". The former is the third moment of returns, and the latter is what volatility traders/portfolio managers usually associate with the difference between two implied vo... |
I applied a smoothing function to a Brownian equation and obtained a stochastic differential equation by using Ito's lemma. The smoothing function is exp(Bt).
How do I get the expected value and variance of this function? Just looking for the required approach rather than a full fledged solution. |
How to write a javascript formula for the time to clear debt on a credit card calculator? |
I discuss the books I mentioned in the comments. They all deal with standard (theoretical) asset pricing (starting with one period utility maximisation and then branch off). Other books like [Björk][1], [Shreve][2] or [Jarrow][3] focus more on time continuous models and derivatives pricing and/or I do not discuss them ... |
I discuss the books I mentioned in the comments. They all deal with standard (theoretical) asset pricing (starting with one period utility maximisation and then branch off). Other books like [Björk][1], [Shreve][2] or [Jarrow][3] focus more on time continuous models and derivatives pricing and/or I do not discuss them ... |
Could anyone show me how to get the second row equation from the first row equation please? For each letter, $p$ is the risk-neutral probability in the risk-neutral world, $u$ is the up factor for the stock, and $d$ is the down factor for the stock, S0 is the beginning stock price. The equation is based on a one-step b... |
I discuss the books I mentioned in the comments. They all deal with standard (theoretical) asset pricing (starting with one period utility maximisation and then branch off). Other books like [Björk][1], [Shreve][2] or [Jarrow][3] focus more on time continuous models and/or derivatives pricing and I do not discuss them ... |
I am currently reading Cochranes book on asset pricing. However, I get confused about one thing. He says that one could test a factor model (I will use the CAPM, just as he does), via a time series approach (a) and a cross section approach (b) (and the Fama McBeth procedure which I do not want to discuss here, though).... |
I am currently reading Cochranes book on asset pricing. However, I get confused about one thing. He says that one could test a factor model (I will use the CAPM, just as he does), via a time series approach (a) and a cross section approach (b) (and the Fama McBeth procedure which I do not want to discuss here, though).... |
I'm taking a course 'AI for trading' in udacity, and there is a part I really want to make sure.
The lecture keeps teaching me that there are alpha factors(driver of return) and risk factors(driver of variance) and it also says that alpha factors can turn into risk factors. I've heard that there is a joke between quan... |
I'm taking a course 'AI for trading' in udacity, and there is a part I really want to make sure.
The lecture keeps teaching me that there are alpha factors (driver of return) and risk factors (driver of variance) and it also says that alpha factors can turn into risk factors. I've heard that there is a joke between qu... |
Assume you have bought a multicallable bond where the issuer has the right to redeem the notional at various dates, e.g. a $10$ yr maturity, $5$% coupon yearly and each year one call date. Next, assume your risk management is not very sophisticated so that it is not able to measure the risk of such "exotic" bonds.
I... |
There are many (hundreds?) of ways to get historical equity data online, both free and paid.
Check out the "What data sources are available online?" megathread: https://quant.stackexchange.com/questions/141/what-data-sources-are-available-online?page=2&tab=votes#tab-top
I have my own data service I just started, ht... |
I have computed the theoretical minimum variance portfolio using the 30 stocks in the Dow.
The formula used is:
$$\underset{N\times 1}{\omega_{mvp}}=\frac{\lambda}{2}\cdot \Sigma^{-1}\iota=\frac{\Sigma^{-1}\iota}{\iota'\Sigma^{-1}\iota}$$
Where $\iota$ is a $N\times 1$ vector containing 1's.
For the data do... |
Can you please confirm on the following?
The difference between TONA (also called TONAR), JPY Libor, TIBOR is that:
- JPY Libor, TIBOR are based on quotes from panel banks. The difference between them is that Libor is the offshore yen rate based on UK banks quotes and TIBOR the onshore yen rate based on Japanes... |
I am trying to understand the connection between market completeness and risk neutral measures.
The market is complete if and only if the martingale measure is unique.
But if I change to the T-Forward measure I will have an equivalent measure to the money market account numeraire measure, so two different but equival... |
If there is a T-Forward measure and a risk neutral measure then markets are not complete? |
When the open interest increases, this means that there is a buyer and a seller of that option.
Both seller and buyer are behooved to hedge their positions, with the opposite sign; but I doubt that both hedge equally aggressively or systematically, with my guess being that it is the seller who hedges more often (be... |
Who hedges (more): options seller or options buyer? |
Why did you expect to hit the minimum variance portfolio in your simulations?
There are infinitely many vectors of size Nx1, such that the sum of its elements adds up to 1. Even with 1 million simulations, it is very difficult to hit the exact minimum variance allocation vector.
Moreover, your simulations might h... |
I have computed the theoretical minimum variance portfolio using the 30 stocks in the Dow.
The formula used is:
$$\underset{N\times 1}{\omega_{mvp}}=\frac{\lambda}{2}\cdot \Sigma^{-1}\iota=\frac{\Sigma^{-1}\iota}{\iota'\Sigma^{-1}\iota}$$
Where $\iota$ is a $N\times 1$ vector containing 1's.
For the data do... |
Does sombody know exactly if a TRS is always breakable ? Or if breaking the TRS position is an option in the term-sheet.
I need an accurate response.
Thank you ! |
Is Breakable TRS subject to an option in the termsheet? |
I am trying to understand the connection between market completeness and risk neutral measures.
A market is complete if and only if the equivalent martingale measure is unique.
But if I change to the $T$-forward measure ($\mathbb{Q}^T$), I have an equivalent measure to the money market account numéraire measure ($\... |
If there is a $T$-forward measure and a risk neutral measure, then markets are not complete? |
Could anyone show me how to prove that the European call option value has a positive relationship with the risk-free rate in a two-step binomial model with strike price K and different risk neutral probability q between each step?
I know that in continuous model, the rho(call) is positive which shows the positive rel... |
I have the following question :
How can €str rate be lower than ecb deposit facility rates (deposit yield) ?
Why would a bank lend money at that rate ?
Thank you for your answers |
It's normal that it takes _very_ long to come close to
the efficient frontier with random portfolios.
How close you come how fast will be strongly influenced
by how you sample the portfolios. In your code, you
sample uniformly. You may want to look at the weight
distributions of the portfolios on the frontier, a... |
I think I understand your issue. Your weights are not uniformly distributed over the desired space. Instead you weights are bunched together, there are fewer extreme weights (weights close to 1) than there should be.
For example let's consider how often the 1st weight (or any other weight for that matter) should be ... |
I have computed the theoretical minimum variance portfolio using the 30 stocks in the Dow.
The formula used is:
$$\underset{N\times 1}{\omega_{mvp}}=\frac{\lambda}{2}\cdot \Sigma^{-1}\iota=\frac{\Sigma^{-1}\iota}{\iota'\Sigma^{-1}\iota}$$
Where $\iota$ is a $N\times 1$ vector containing 1's.
For the data do... |
In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$
Here $\mu$ and $\sigma$ are constants, $W_{t}$ is a Wiener process, $N_{t}$ is a Poisson process, and $Y_{i}$s are $N\left(... |
Nothwithstanding @oliversm's excellent response to my question, I think there's also a question on how to interpret $\sigma(t,S(t))$ in a local vol model.
Given that $\sigma(t,S(t))$ comes from a Markovian projection of a complicated stochastic process on to a simpler one, my interpretation (perhaps wrong) is that ... |
I have the following question :
How can €str rate be lower than ecb deposit facility rates (deposit yield) ?
Why would a bank lend money at that rate ?
Thank you for your answers
Edit : for instance Eonia rates are always higher than deposit yield, but not €str as it seems which is lower since july |
Is there a way to calculate the yield of a bond knowing the DV01 and duration (numerical values) and knowing all other parameters of $V$?
$$DV01=-\frac{\partial V}{\partial y}$$ |
Quantlib: Why Interest Rate Swap valuation throwing the 'Runtime error negative time'? |
Market-makers gain profit from the bid-ask spread. This means that they could place speculative positions without any cost (they pay the spread to themselves = zero cost). I assume there are laws preventing it because if it could happen, the opportunities are far from equal for each speculative trader (I have to overco... |
What prevents market-makers to do speculative trades with much better conditions than other traders? |
Assuming that the stock has 100% annual turnover, this amounts to buying 2.5 days of total volume. Of course, such a trade will take a long time to execute.
The question is: suppose the trade can be done very slowly. What kind of long-term market impact would be predicted by typical transaction cost models on Wall St... |
What do typical transaction cost models predict about the long-term price impact of buying 1% of a stock’s shares outstanding? |
Not sure about OTC, but on most exchanges cross-trades where buyer and seller is the same counter-party are not allowed. A market maker that trades with itself would be subject to the same rules, and besides such trades would still incur exchange commissions. Actually the exchanges typically pay market makers to be pre... |
Not sure about OTC, but on most exchanges cross-trades where buyer and seller is the same counter-party are not allowed. A market maker that trades with itself would be subject to the same rules, and besides such trades would still incur exchange commissions. Actually the exchanges typically pay market makers [1] to be... |
Is there a way to calculate the yield of a bond knowing the DV01 and duration (numerical values) and knowing all other parameters of $V$ (maturity, coupon, etc.)?
$$DV01=-\frac{\partial V}{\partial y}$$ |
I'm new to this field but I'm reading related literature lately and quite obsessed with the topic. I come to know that people like to model short rate under risk-neutral measure $Q$, because under $Q$ the price of a zero-coupon bond can be expressed as $$P(t,T)=\mathbb{E}_Q\left[e^{-\int_t^Tr_sds}|\mathscr{F}_t\right],... |
I'm new to this field but I'm reading related literature lately and quite obsessed with the topic. I come to know that people like to model short rate under risk-neutral measure $Q$, because under $Q$ the price of a zero-coupon bond can be expressed as $$P(t,T)=\mathbb{E}_Q\left[e^{-\int_t^Tr_sds}|\mathscr{F}_t\right],... |
**This post is Quant Stack Exchange's master list of data sources.**
Please append your links to other data sources to the list below.
## Economic Data ##
<br>
See [What are the most useful sources of economics data?][1] on Cross Validated.
**World**
- [OECD.StatExtracts][2] includes data and metadata fo... |
I have a question about merging two sets of timeseries without causing much havoc.
I have to merge dataset 1 into dataset 2.
Problem is:
1. The levels are different. (Ex: dataset 1 is moving between 2 - 3 while dataset 2 is moving between 5 - 6)
2. I have to maintain the same consistency in the returns from datas... |
How to merge two sets of timeseries together while maintaining the same returns? |
In the world of Quantitative `model risk management`, can you please tell me what is `Model 1` type, `Model 2` type etc? |
Does anyone have a good reference on how to derive time weighted vega for options? The only literature I found was in this presentation:
http://www.topquants.nl/wordpress/wp-content/uploads/2015/01/Van-Gulik-Risk-management-at-Optiver.pdf
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.... |
Does anyone have a good reference on how to derive time weighted vega for options? The only literature I found was in this presentation:
http://www.topquants.nl/wordpress/wp-content/uploads/2015/01/Van-Gulik-Risk-management-at-Optiver.pdf
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.... |
I constructed the local volatility surface of S&P 500 from implied vols and was able to price the options accurately using Monte-Carlo. Let's say I priced a 80% of S0 put option with S0 = 4000. How do I approach the MonteCarlo pricing of this 80% put option if my S moves to 4100 instantaneously? My question is I want ... |
I constructed the local volatility surface of S&P 500 from implied vols and was able to price the options accurately using Monte-Carlo. Let's say I priced a 80% of S0 put option with S0 = 4000. How do I approach the MonteCarlo pricing of this 80% put option if my S moves to 4100 instantaneously? My question is I want ... |
i am looking for a simple parametric VaR approach that includes vega and/or gamma.
i want to choose a non symmetric distribution for the pnl, what would people use, perhaps some shifted lognormal (shifted so it's mean is zero) ?
also if anyone has any links to maths that show the derivation for the isda simm formu... |
what is a simple parametric VaR approach that can be used to compare with ISDA SIMM results? |
First of all, similar questions like mine are answered on this forum but I never quite saw an answer to this specific question. I'm trying to predict inflation by using an AR model with exogenous variables, or ARX. I know how to figure out the order for the AR part, but it is not clear to me how to pick the exogenous v... |
I am writing my Master Thesis on momentum strategies including price momentum (UMD/MOM) and two fundamental momentum strategies (SUE and CAR3). Right now I'm trying to create the three momentum factors in order to compare their performance from 1975-2012. The UMD/MOM factor is a factor already constructed by Fama-Frenc... |
i am looking for a simple parametric VaR approach that includes vega and/or gamma.
i am not sure if to choose a non symmetric distribution for the pnl, what would people use, perhaps some shifted lognormal (shifted so it's mean is zero) ? , or to just choose normal. i think isda simm uses normal , but then, how do t... |
>**Problem**: Consider a new derivative that at time $T$ pays $Y =(S(T_0) − S(T))^+$
>where
$0 < T_0 < T$ is a fixed date.
>(i) Show that the arbitrage-free of Y at time $t = T_0$ is given by $\pi_{T_0}
(Y ) =pS(T_0)$ where p is independent of the stock price.
>(ii) Determine the arbitrage-free price of deri... |
Firstly I should mention that I am new to both Eviews and GARCH models.
Anyway, I am conducting some research into the effect that different macroeconomic factors have had on stock index volatility from 2009 until 2020. Is there any way to include these macroeconomic time series when estimating the GARCH equation?
... |
If we follow the mean-risk ptf optimization strategy and use variance/VaR/ES as risk measure, the results in general are different. In other words the **efficient frontier are not shared**. However, as already said by Enrico Schumann, if the multivariate returns distribution of the assets is *Elliptical* (with finite v... |
$WACC = \frac{E}{V} * R_e + \frac{D}{V} *R_D(1-T_c) $
where $R_D$ should be the cost of the debt. $T_c$ is the tax rate.
When googling the formula for cost of debt I find $Cost of debt = R_D = Interest*(1-tax rate)$
But this looks identical to the $R_D(1-T_c$) part, if only $R_d$ was interest rate in the WACC Fo... |
It is matter of choice of model
- do you believe that the volatility of each of your stock / factors is stationary?
- what is, according to you, the needed number of observation to compute the coef of your covariance matrix with accuracy?
For instance, results on [rough volatility][1] suggest that if you want to... |
MES refers to Micro E-mini S&P 500 Index Futures
As my understanding of futures goes, future price increases with contract end time because otherwise, one can buy the longer expiring contract and sell the short expiring future, then start shorting the underlying assets of the index to achieve arbitrage.
However, ... |
Why MES2106 is having a lower price than MES2103? |
It is sometimes said that “pension and insurance hedgers push the long end of the curve down”
Explain how this happens and why- what are the connected steps?
What do these players do and why?
How does this result in curve moves?
Also – how does this activity affect inflation curves? |
**Is GBM still viable for modeling asset prices in a very short time period ?** I assume by asset price you mean the mid-price of the asset. Ito diffusions are unable to capture stylized facts of market microstructure: in particular, you get volatility clustering, negligible autocorrelation in return series, the signa... |
I am trying to understand how to calculate the P&L on a eurodollar futures options position. Suppose I am looking at say Dec-2023 99.125 strike put options with a bid-ask of 0.2250 - 0.415.
Since this is not a front month contract CME says that the minimum price fluctuation is one half of one basis point (0.005 = $1... |
**Is GBM still viable for modeling asset prices in a very short time period ?** I assume by asset price you mean the mid-price of the asset. Ito diffusions are unable to capture stylized facts of market microstructure: in particular, you can't get volatility clustering, negligible autocorrelation in return series, the... |
It will be extremely hard for anyone to follow your computations without the data you have used. But to venture a guess: In the figure in the Novy-Marx paper, the series are scaled to a vol of 10%. What is the volatility of your series? |
I am writing my Master Thesis on momentum strategies including price momentum (UMD/MOM) and two fundamental momentum strategies (SUE and CAR3). Right now I'm trying to create the three momentum factors in order to compare their performance from 1975-2012. The UMD/MOM factor is a factor already constructed by Fama-Frenc... |
Apologies for sounding patronising as hell if this isn't the problem... but do the dates associated with your EPS data refer to (1) the accounting period in question, or (2) the dates these were reported?
So if (say) GE or JPM was reporting calendar Q4 2020 on Valentine's Day, would this be 31/12/2020 or 14/02/2021... |
This is an issue for legacy Defined Benefit pension funds, that promise to pay a fraction of salary for time worked. Virtually nobody offers these schemes these days; but enough (usually older, and often public-sector) workers and retirees still have related entitlements thus.
These assets to the employee are obvio... |
I've just (hypothetically) invested $1m in two market-cap-weighted global equities tracker funds: one large+mid-cap tracker and one small-cap tracker. Both are of an accumulation share class, and I balanced my investment by respective market caps.
I'm trying to understand the underlying theory and if/why I need to r... |
I've just (hypothetically) invested $1m in two market-cap-weighted global equities tracker funds: one large+mid-cap tracker and one small-cap tracker. Both are of an accumulation share class, and I balanced my investment by respective market caps.
I'm trying to understand the underlying theory and if/why I need to r... |
Usually carry trades involve borrowing in a low yield currency and invest in a high yield currency. For example, I borrow dollars and invest in Brazilian gov bonds. I then use a rolling FX swap to hedge the FX risk (eg sell dollars spot and buy them back forward).
I recently read about a different carry trade on ... |
Usually carry trades involve borrowing in a low yield currency and invest in a high yield currency. For example, I borrow dollars and invest in Brazilian real. I then use a rolling FX swap to hedge the FX risk (eg sell dollars spot and buy them back forward).
I recently read about a different carry trade on the M... |
**Is GBM still viable for modeling asset prices in a very short time period ?** I assume by asset price you mean the mid-price of the asset. Ito diffusions are unable to capture stylized facts of market microstructure: in particular, you can't get volatility clustering, negligible autocorrelation in return series, the... |
I wonder why sellers of Autocalls should cover the gamma ?
The autocall I'm talking about is of this type :
While the spot never goes below 70 % of the initial value, nothing happens.
If it happens, then the payoff becomes a short put with strike 100 %.
The seller is never at risk, except from a marked to m... |
Why should a seller of autocall (down out ones) cover gamma risk? |
Market participants use negative interbank rates (LIBOR JPY/CHF) for the valuation of FX futures. Does this make any economic sense?
Positive rates in valuation formula indicate opportunity cost of money, but since rates are negative, are there any "opportunity" to be accounted for?
Why not just set negative rates... |
Market participants use negative interbank rates (LIBOR JPY/CHF) for the valuation of FX futures. Does this make any economic sense?
Positive rates in valuation formula indicate opportunity cost of money, but since rates are negative, are there any "opportunity" to be accounted for?
Naive way to ask: Why not just ... |
In an american callable bond there is an expectation for the issuer to prepay its debt prior to maturity. I understand that this reduces it's value and therefore, higher yield.
But another way to think about it is that investing in a callable bond should yield the same as a non-callable bond with lower duration (due... |
Why do a callable bond always have higher yields? |
I am solving a problem in which following payoff is provided:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/1nsFG.jpg
With $S_0=100$ and $T=8$. Looking at the payoff it seems obvious that it is replicated with two european put options ($K=100$ and $K=150$) and an european call option... |
I am solving a problem in which following payoff is provided:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/1nsFG.jpg
With $S_0=100$ and $T=8$. Looking at the payoff it seems obvious that it is replicated with two european put options ($K=100$ and $K=150$) and an european call option... |
[![enter image description here][1]][1]
The above chart plots a 300 weekly exponential average. With 52 weeks per year, the indicator would start calculating the EMAVG after 5.77 years, in 2016.
What technique could it be using to be able to calculate EMAVG values for the weeks before 2016?
[1]: https://i.st... |
Exponential moving average values before the data range is met? |
I am solving a problem in which following payoff is provided:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/1nsFG.jpg
With $S_0=100$ and $T=8$. Looking at the payoff it seems obvious that it is replicated with two european put options ($K=100$ and $K=150$) and an european call option... |
The exponential weighting scheme yields an estimate $Z_t$ from observations $X_t$
$$
Z_t=\sum_{k=0}^{\infty} w_k X_{i-k}
$$
where the (infinite) series of weights sums to one, i.e.
$\sum_{k=0}^{\infty} w_k=1$.
Let's call the exponential weighting parameter $\lambda$. For a 'lookback window' of length $N$ el... |
The exponential weighting scheme yields an estimate $Z_t$ from observations $X_t$
$$
Z_t=\sum_{k=0}^{\infty} w_k X_{t-k}
$$
where the (infinite) series of weights sums to one, i.e.
$\sum_{k=0}^{\infty} w_k=1$.
Let's call the exponential weighting parameter $\lambda$. For a 'lookback window' of length $N$ el... |
I am solving a problem in which following payoff is provided:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/1nsFG.jpg
With $S_0=100$ and $T=8$. Looking at the payoff it seems obvious that it is replicated with two european put options ($K=100$ and $K=150$) and an european call option... |
Generally speaking, when calibrating a local volatility model a la Dupire to European vanilla calls, should I use the numerically (PDE or Monte Carlo) solved price for the vanilla call in the cost function or is there an analytical formula for vanillas in local volatility model I could use to speed up the optimization ... |
I am solving a problem in which following payoff is provided:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/1nsFG.jpg
With $S_0=100$ and $T=8$. Looking at the payoff it seems obvious that it is replicated with two european put options ($K=100$ and $K=150$) and an european call option... |
*"Would you be able to explain the rationale behind this [buy spot...enter a FX swap where you sell spot and buy it back forward]?"*
Very few accounts want to take delivery of a spot position therefore they roll via the swap. |
try to be clear to ask my question:
Suppose the original vol surface is a n by m matrix where n is the number of pillars in the volatility term structure and m is the number of strikes. According to my portfolio of options with different strikes and maturities, suppose my vega is distributed along the original n by m ... |
How could option vega be remapped on reduced volatility surface? |
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